ESI The Erwin Schrodinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, Austria

Exploiting N=2 in Consistent Coset Reductionsof Type IIA

Davide Cassani

Amir-Kian Kashani-Poor

Vienna, Preprint ESI 2110 (2009) February 11, 2009

Supported by the Austrian Federal Ministry of Education, Science and CultureAvailable via http://www.esi.ac.at

IHES/P/09/03LPTENS-09/02

ROM2F/2009/01January 2009

Exploiting N = 2 in consistent coset

reductions of type IIA

Davide Cassania,b∗ and Amir-Kian Kashani-Poor c†

a Laboratoire de Physique Theorique‡, Ecole Normale Superieure,24 rue Lhomond, 75231 Paris Cedex 05, France

b Dipartimento di Fisica, Universita di Roma “Tor Vergata”Via della Ricerca Scientifica, 00133 Roma, Italy

c Institut des Hautes Etudes ScientifiquesLe Bois-Marie, 35, route de Chartres

91440 Bures-sur-Yvette, France

Abstract

We study compactifications of type IIA supergravity on cosets exhibiting SU(3)structure. We establish the consistency of the truncation based on left-invariance,providing a justification for the choice of expansion forms which yields gauged N = 2

supergravity in 4 dimensions. We explore N = 1 solutions of these theories, empha-sizing the requirements of flux quantization, as well as their non-supersymmetric

companions. In particular, we obtain a no-go result for de Sitter solutions at stringtree level, and, exploiting the enhanced leverage of the N = 2 setup, provide a

preliminary analysis of the existence of de Sitter vacua at all string loop order.

∗cassani AT lpt.ens.fr†kashani AT ihes.fr‡Unite mixte du CNRS et de l’Ecole Normale Superieure associee a l’Universite Pierre et Marie Curie

Paris 6, UMR 8549.

1 Introduction

In the era of LHC, much effort is being invested in finding phenomenologically viable stringvacua. Much of this work takes place by considering compactifications to N = 1 theoriesin 4d. In this paper, we will focus instead on a framework which yields 4 dimensionaltheories that have N = 2 symmetry realized off-shell. While the N = 1 setup allows formore flexibility in choosing the various ingredients of the theory, and hence (currently)permits the construction of more realistic vacua, the increased rigidity of the N = 2setup has the advantage of allowing a more exhaustive treatment of α′, string loop, andforeseeably even brane instanton corrections. An impressive example of the power of theN = 2 framework is the recent proof [1] that N = 2 gauged supergravities without vectormultiplets do not permit de Sitter vacua, in spite of the presence of such solutions in theone-brane-instanton approximation [2]. Studying theories in the N = 2 framework hencepresents one promising avenue towards assessing the viability of the approximations thatare necessary to get off the ground in less supersymmetric frameworks.

The best studied example of N = 2 theories obtained from string theory are type IICalabi-Yau compactifications [3, 4]. The differential operators governing the geometricmoduli problem of the internal Calabi-Yau manifolds turn out to coincide with the massoperators of the supergravity theory. Unobstructed deformations hence give rise to mass-less excitations, resulting in the beautiful identification between the massless scalar fieldsof these theories, whose VEVs parametrize a family of supergravity solutions, and thegeometric moduli of the Calabi-Yau. The masslessness of the scalars is protected by su-persymmetry, as N = 2 forbids a potential in the case of uncharged matter. In [5], thestudy of type II compactifications on SU(3) structure manifolds was initiated (recall thatCalabi-Yau manifolds satisfy the stronger condition of SU(3) holonomy). This setup ismore akin to the phenomenologically motivated N = 1 analyses: solutions of the super-gravity equations of motion on these internal manifolds require the presence of backgroundfluxes [6, 7, 8], and compactification gives rise to 4d N = 2 gauged supergravity theories[9, 10], which, in contrast to the Calabi-Yau case with uncharged matter, exhibit a poten-tial for the scalar fields in the theory. The increased phenomenological viability comes ata price: the very presence of a potential makes it unlikely that the choice of light degreesof freedom of the theory can be associated to a geometric moduli problem. Indeed, asystematic approach to a reduction ansatz for these theories is still lacking. Following ourwork in [11] and [12], we here pursue an alternative approach towards justifying the reduc-tion ansatz, that of consistent truncation: obtaining a field theory with a finite numberof fields upon compactification requires truncating most of the degrees of freedom of thehigher dimensional theory; this truncation is called consistent when all solutions to thelower dimensional equations of motion lift to solutions of the higher dimensional theory.Note the contrast to a Kaluza-Klein reduction [13], which is an expansion valid around asingle 10d solution (hence referred to as a base-point dependent reduction in [14]).

Consistently truncated lower dimensional field theories are powerful allies in studying thevacuum structure of the higher dimensional string theory. This is partially a consequence

1

of computational techniques being more refined in lower dimensions. E.g., various leadingnon-trivial contributions in α′ to the 10d type II supergravity action have been determined[15, 16, 17, 18]. One may hope to establish the complete action to this order by 10dsupersymmetric completion [19]. However, the 10d supersymmetry equations have simplyproved too cumbersome to date. By contrast, the supersymmetric completion of thecontribution of these terms to the 4d N = 2 supergravity action is readily available,yielding the full string tree level and one loop corrected action. In fact, in 4d we can,as we will discuss, even draw conclusions regarding the all string loop corrected action.Studying the lower dimensional theory is however not merely a question of computationalconvenience. An effective higher dimensional description of worldsheet or brane instantonsis even conceptually problematic.

In [12], it was shown that expansion forms can be defined on Nearly Kahler manifolds thatsatisfy the conditions of [14], implying that the reduction of the type IIA action based onthese forms yields N = 2 gauged supergravity in 4d. It was further demonstrated thatthe truncation in this setting is consistent in the supersymmetric sector (i.e. 4d solutionspreserving N = 1 supersymmetry lift). In this paper, we shift our focus to certaincoset spaces which subsume the currently known set of 6d Nearly Kahler manifolds. Weintroduce these spaces in section 2. Considering the emphasis on base point independenceof the reduction, it was perhaps somewhat disappointing that the theories based on NearlyKahler reduction yielded a single supersymmetric vacuum for a given choice of fluxes.Cosets by contrast permit multiple N = 1 solutions for a given choice, which are allaccessible via the 4d theory. We demonstrate this in section 3. Due to flux quantization,the solutions come in a discrete family. We perform the required K-theory analysis. Insection 4, we demonstrate that the left-invariant coset reductions represent a consistenttruncation by establishing that the 10d equations of motion reduce to the 4d equationsfollowing from the appropriate N = 2 action. This extends the analysis of [11] beyondthe RR sector and overcomes the restriction to consistency merely of the supersymmetricsector [12, 20]. Fueled by this result, we turn to the study of non-supersymmetric vacuaof the 4d theories in sections 5 and 6. We find several non-supersymmetric Nearly Kahlercompanions to the solution of section 3 and study their stability, in particular with regardto deformations away from the Nearly Kahler locus. We also consider the question of theexistence of de Sitter vacua, which has received some attention recently in the type IIAcontext [21, 22, 1, 23, 24, 25]. We demonstrate that such vacua are absent at string treelevel (we prove this result in greater generality than the coset context: it is valid for anygauged supergravity with merely the universal tree-level hypermultiplet, irrespective ofthe specifics of the vector multiplet sector). Due to the increased leverage in the N = 2setup, we are able to push this analysis beyond tree level. We obtain the full string loopcorrected potential, which evades the tree-level no-go theorem, and uncover a necessarycondition on the contribution of the NSNS sector to the potential for de Sitter vacua tobe possible. In two appendices, we fill in the details of the dimensional reduction leadingto the 4d N = 2 theory (appendix A), and study the string loop corrected 4d N = 1conditions (appendix B).

2

2 Introducing the internal geometries

We consider dimensional reductions of massive type IIA supergravity on left coset spacesM6 = G/H endowed with a left-invariant SU(3) structure. An exhaustive list of suchcosets was provided in ref. [26] (see section 1 and in particular table 1 therein). In thefollowing, we are going to focus on the cosets whose SU(3) structure cannot be furtherreduced to SU(2), namely

SU(3)

U(1) ×U(1),

Sp(2)

S(U(2) ×U(1)),

G2

SU(3), (2.1)

where S(U(2) × U(1)) is non-maximally embedded in Sp(2).

It is easy to see that a reduction performed on these manifolds by expanding the higherdimensional fields in a basis of left-invariant forms satisfies the constraints of [14] andtherefore yields a gauged N = 2 supergravity in 4d.

The remaining cosets listed in [26] have vanishing Euler characteristic and admit a left-invariant vector: their SU(3) structure group is therefore further reduced to at least SU(2).For these cosets, the N = 2 reduction ansatz based on the presence of SU(3) structurecan be more naturally enlarged to include the whole set of left-invariant forms, possiblyyielding a further extended supergravity (N ≥ 4) in 4d.

The only non-vanishing torsion classes1 characterizing the SU(3) structure of the cosets(2.1) are W1 and W2, i.e. the SU(3) invariant 2- and 3-form J and Ω satisfy

dJ =3

2Im(W1Ω) ,

dΩ = W1J ∧ J + W2 ∧ J . (2.2)

In fact, G2

SU(3)allows just W1 6= 0 and is therefore a Nearly Kahler manifold. The cosets

SU(3)U(1)×U(1)

and Sp(2)S(U(2)×U(1))

also admit a region in the SU(3) structure parameter space inwhich they are Nearly Kahler, but in general, their W2 torsion class does not vanish. SinceW1 and W2 can be chosen purely imaginary, these cosets fall into the class of ‘half-flat’manifolds, characterized by Re W1 = Re W2 = W4 = W5 = 0 [28].

A description of the coset spaces (2.1) was given e.g. in [29]. In the context of SU(3)structure compactifications of (massive) type IIA supergravity, supersymmetric AdS4

backgrounds on these manifolds have recently been found in [26, 30, 31, 32] and fur-ther discussed in [33], while refs. [34, 23] study the properties of the associated effective4d N = 1 supergravity in the presence of orientifold projections (see also [31] for a previ-

ous work considering the coset SU(3)U(1)×U(1)

). Type IIA reduction on Nearly Kahler manifolds

has been worked out in [12]. The cosets (2.1) first appeared in the string literature in[35, 36] in the heterotic context, and have also been employed recently in [37] for heteroticdimensional reductions.

1For a review of SU(3) structures and their torsion classes, see e.g. subsection 3.2 of ref. [27].

3

2.1 The expansion forms

In the following we provide the most general left-invariant positive-definite metric for eachcoset (2.1), as well as a basis for all the left-invariant differential forms, on which we aregoing to expand the supergravity fields.

We define the 6d coset spaces (2.1) as in ref. [26], and in particular adopt the set of groupstructure constants listed therein. The same reference also provides a summary of theneeded mathematical notions about coset spaces, while a more extended review can befound e.g. in [29].

Using the local coframe2 em inherited from G, a differential form on the coset G/Hreads ωk = 1

k!ωm

1...,m

kem

1 ∧ · · · ∧ emk . This is invariant under the left action of G if

its components are constant and satisfy the following relation involving the G structureconstants

fpi[m

1ωm

2...,m

k]p = 0 , (2.3)

where the index i is associated with the generators of the algebra h, while the underlinedindices label a basis for the complement of h in g. For the coset metric ds2 = gmnem ⊗ en

the relation is analogous to (2.3), with a symmetrization of indices replacing the antisym-metrization. The action of the exterior derivative preserves left-invariance, and is alsodetermined by the structure constants of G.

None of the cosets we consider admits left-invariant 1– or 5–forms.

We define the ‘standard volume’ of the cosets as

I :=

∫

e123456 .

2.1.1 SU(3)U(1)×U(1)

Left-invariant metric:

gmn = diag(v1, v1, v2, v2, v3, v3) , v1 > 0, v2 > 0, v3 > 0 . (2.4)

The left-invariant forms are spanned by

ω0 = 1 , ω1 = −e12 , ω2 = e34 , ω3 = −e56 ,

α =1

2√

I(e135 + e146 − e236 + e245) , β =

1

2√

I(−e136 + e145 − e235 − e246) ,

ω0 =1

Ie123456 , ω1 =

1

Ie3456 , ω2 = −1

Ie1256 , ω3 =

1

Ie1234 . (2.5)

2Here and in the following (see in particular subsection 4.3), frame indices are underlined.

4

2.1.2 Sp(2)S(U(2)×U(1))

Left-invariant metric:

gmn = diag(v1, v1, v1, v1, v2, v2) , v1 > 0, v2 > 0 . (2.6)

Basis of left-invariant forms:

ω0 = 1 , ω1 = −e12 − e34 , ω2 = e56 ,

α =1

2√

I(e135 + e146 + e236 − e245) , β =

1

2√

I(e136 − e145 − e235 − e246) ,

ω0 =1

Ie123456 , ω1 =

1

2I(e1256 + e3456) , ω2 = −1

Ie1234 . (2.7)

2.1.3 G2

SU(3)

Left-invariant metric:

gmn = diag(v1, v1, v1, v1, v1, v1) , v1 > 0 . (2.8)

Basis of left-invariant forms:

ω0 = 1 , ω1 = −e12 + e34 − e56 ,

α =1

2√

I(e135 + e146 − e236 + e245) , β =

1

2√

I(−e136 + e145 − e235 − e246) ,

ω0 =1

Ie123456 , ω1 =

1

3I(e3456 − e1256 + e1234) . (2.9)

2.1.4 Properties

The overall factors in the basis forms (2.5), (2.7), and (2.9) have been chosen in such away that

∫

〈ωA, ωB〉 = δBA ,

∫

α ∧ β = 1 , (2.10)

where A = (0, a) , B = (0, b) and a, b label the left-invariant 2– and 4–forms. The an-tisymmetric pairing 〈 , 〉 is defined on even forms ρ, σ as 〈ρ, σ〉 = [λ(ρ) ∧ σ]top, with

λ(ρk) = (−)k

2 ρk , k being the degree of ρ.

The basis forms define a closed differential system,

dωa = qaα ,

dα = 0 , dβ = qaωa ,

dωA = 0 , (2.11)

which is also closed under the action of the Hodge star operator,

∗α = β , ∗ω0 =1

Vol, ∗ωa = − 1

4VolGabωb .

5

SU(3)U(1)×U(1)

Sp(2)S(U(2)×U(1))

G2

SU(3)

range of a : 1, 2, 3 1, 2 1

geometric flux qa : q1 = q2 = q3 = −√

I q1 = 2√

I , q2 =√

I q1 = 2√

3I

Gab = diag(

4(v1)2 , 4(v2)2 , 4(v3)2)

diag(

2(v1)2 , 4(v2)2)

43(v1)2

Vol = v1v2v3I (v1)2v2I (v1)3I

I = 25π3 27π3

3144π3

5

Table 1: Values of the different quantities introduced in subsection 2.1.

Here, the qa encode what are sometimes referred to as geometric fluxes, Vol denotes thevolume of the coset, and the matrix Gab is the inverse of

Gab =1

4Vol

∫

ωa ∧ ∗ωb , (2.12)

corresponding to the special Kahler metric on the space of the internal metric and B-fielddeformations [14]; see subsection A.1 of the appendix for more details.

In table 1, we give the values of the quantities introduced above for each coset. Thestandard volume I was computed following ref. [29].3 Its evaluation requires knowledgeof the Euler characteristic of our cosets. Since the harmonic forms on a compact cosetreside among the left-invariant forms, we can read off the cohomology from the differentialrelations (2.11). We immediately conclude that all our cosets have trivial odd cohomology.

Concerning the even cohomology, for SU(3)U(1)×U(1)

, with

ω′1 = ω1 − ω3 , ω′

2 = ω2 − ω3 , (2.13)

we haveH2 = Span ([ω′

1], [ω′2]) , H4 = Span

(

[ω1], [ω2])

,

hence the Euler characteristic is χ = 6.

For Sp(2)S(U(2)×U(1))

, we have b2 = 1 and χ = 4, while for G2

SU(3), b2 = 0 and χ = 2.

2.2 The SU(3) structure

For each coset in (2.1), the pair of left-invariant forms parametrized by va,

J = vaωa , Ω = 2√

Vol(α + iβ) , (2.14)

3We have a 26 supplementary factor in I with respect to [29]. This is due to the fact that for thenormalization of the group structure constants we follow the choice of [26], and this differs from the oneof [29] by a factor 1/2.

6

satisfies the relations J ∧ Ω = 0 and 3i4Ω ∧ Ω = J ∧ J ∧ J and hence determines a left-

invariant SU(3) structure. The metric specified by J and Ω is precisely the one given ineq. (2.4), (2.6), and (2.8) respectively for the three cosets. Using the properties of thebasis forms listed in subsection 2.1.4 above, one can see that the differential relations (2.2)are satisfied, with torsion classes4

W1 = − ivaqa

3√

Vol, (2.15)

W2 = − 2i

3√

Volqa

(

vavb − 3

4Gab)

ωb.

Substituting the quantities given in the table of subsection 2.1.4, we see that the NearlyKahler condition W2 = 0 is identically satisfied on G2

SU(3). For Sp(2)

S(U(2)×U(1))and SU(3)

U(1)×U(1),

this condition is satisfied on a line in the parameter space determined by v1 = v2 andv1 = v2 = v3 respectively. In this Nearly Kahler limit the cosets are Einstein manifolds(the only other loci at which the Einstein condition is satisfied are 2v1 = v2 for Sp(2)

S(U(2)×U(1))

and 2v1 = 2v2 = v3, or cyclic permutations of this, for SU(3)U(1)×U(1)

[29] ).

The forms (2.14) are the most general left-invariant pair satisfying the SU(3) structuredefining relations (the overall phase of Ω is unphysical; requiring the torsion classes to bepurely imaginary, as we have done, fixes it up to a sign). In particular, since the volumeVol is fixed by the va, we see that Ω identifies a rigid SL(3,C) structure, and there are noalmost complex structure moduli.

2.3 An alternative basis?

In [14], conditions on the expansion forms were emphasized that arise when these aremoduli dependent, as is the case with the basis of harmonic forms on which Calabi-Yaureductions are based (the *-ed conditions in section 2 of [14]). For the set of expansionforms that we have introduced above, these conditions are trivially satisfied, as the formsare moduli independent. In this sense, our expansion ansatz here is technically simplerthan in the Calabi-Yau case. However, in a small flux approximation, the laplacian∆ = − ∗ d ∗ d − d ∗ d∗ becomes the mass operator for the modes of the 10d supergravityfields, and an expansion in eigenforms of it is physically motivated. Can we replace theforms introduced above by such a basis of eigenforms?

In the Nearly Kahler case the expansion in eigenforms of the laplacian is further motivatedby the fact that both J and Ω are themselves eigenforms of ∆ [12]. In the more generalcase W2 6= 0, this is still true for Ω,5

∆Ω =(

3|W1|2 +1

4W2yW2

)

Ω , (2.16)

4The evaluation of W2 is performed rewriting the second line of (2.2) as W2 = 2W1J − ∗dΩ.5One needs the relation dW2 = i

4(W2yW2)ReΩ, satisfied by the cosets (2.1).

7

but not for J , which instead satisfies

∆J = 3|W1|2J − 3

2Re (W1W2) .

Considering e.g. the coset SU(3)U(1)×U(1)

, a change of basis sending the 2–forms introduced in

(2.5) to a set of eigenforms of the laplacian is

ω′1 = ω1 − ω3 , ω′

2 = ω2 − ω3 , ω′3 =

∑

a(va)2ωa

∑

b(vb)2

, (2.17)

where ∆ω′1 = ∆ω′

2 = 0 , while ∆ω′3 = (v1)2+(v2)2+(v3)2

v1v2v3 ω′3 . The harmonic 4–forms are

spanned by

∗ω′1 ∝ v3

v1ω1 − v1

v3ω3 , ∗ω′

2 ∝ v3

v2ω2 − v2

v3ω3 , (2.18)

while ∗ω′3 ∝ −

√I(ω1 + ω2 + ω3) = dβ is exact.

The condition va∂vbωa (*7 of [14]) gives rise to a complicated set of equations for possibleva dependent normalization factors of the primed basis. However, it is easy to see uponinspection that the moduli independence of the triple intersection product (condition *8of [14]) cannot be satisfied for any such choice. The question whether the choice of left-invariant expansion forms can be motivated from a Kaluza-Klein reduction point of viewhence remains an interesting open question.

3 Supersymmetric 10d solutions parametrized by fluxes

In this section, we will rewrite the family of N = 1 solutions of the 10d supergravityequations found in [26] in a manner which makes the discreteness of this family as aresult of flux quantization manifest. By [11] and [12], these solutions can be recoveredfrom the 4d point of view. After proving the full consistency of our reduction in section4, we will proceed to complement these solutions with their non-supersymmetric relativesin section 6.

3.1 Flux quantization and K-theory

RR-fields are classified topologically by K-theory classes [38, 39]. This has two conse-quences for the choice of fluxes associated to the RR-fieldstrengths. Firstly, the naiveinteger quantization of fluxes must be replaced by quantization in multiples of fractionsdetermined also by the topology of the compactification manifold. Secondly, not everychoice of flux number satisfying these quantization conditions will possess a K-theory liftand hence be permissible. We will now study these two points in turn.

In [38], fluxes were conjectured to take values in the image of the map√

A(X) ch(·) : K(X) → Heven(X, Q) .

8

ch(x) is the Chern character as extended to a K-theory element x = E − F via ch(x) =ch(E) − ch(F ). Hence,

[F (x)]

2π=

√

A ch(x) , (3.1)

where F =∑5

i=0 F2i denotes a formal sum of all RR-fieldstrengths, and [·] indicatesrational cohomology class (rational rather than integral due to the fractional coefficientsof Chern classes that appear in the expansion of the Chern character). When H 6= 0, theequations of motion and Bianchi identity of F are modified from the naive Maxwell form,enforcing harmonicity of F , to a version of these equations twisted by H. In particular, Fnow satisfies (d−H)F = 0. When H is exact, as will be the case in our study, H-twistedcohomology maps to ordinary cohomology via F → e−BF , where H = dB. It henceproves convenient to introduce a basis of RR fields given by G = e−BF . Equation (3.1)then holds for G rather than F , and the term ‘fluxes’ refers to the cohomology classes[G].

To decide which fluxes we can choose as boundary conditions of our physical system (andthen parametrize our solutions by this choice), we need to decide on electric vs. magneticvariables. Ignoring subtleties related to torsion, which does not enter in a supergravityanalysis, we can choose the electric basis to lie in ⊕3

i=1H2i(X, Q).

Let us now consider the question of flux quantization. To this end, we expand the righthand side of (3.1) in terms of Chern classes for x the class of a vector bundle F on X,

ch0(F ) = rank(F ) , ch1(F ) = c1(F ) , ch2(F ) =1

2[c1(F )2 − 2c2(F )] ,

ch3(F ) =1

3![c1(F )3 − 3c1(F )c2(F ) + 3c3(F )] ,

A = 1 − p1

24+ . . . .

Hence,

[G0]

2π= rank(F ) ,

[G2]

2π= c1(F ) ,

[G4]

2π=

1

2[c1(F )2 − 2c2(F )] ,

[G6]

2π=

1

3![c1(F )3 − 3c1(F )c2(F ) + 3c3(F )]− p1(X)

48c1(F ) .

As Chern classes take value in integral cohomology, it follows that in the presence ofG2 flux, G4/2π is generically half-integrally quantized. Neglecting gravitational effects,G6/2π is quantized in multiples of 1

6, incorporating the A-genus generically yields quanti-

zation in multiples of 148

. In particular, for the cosets we are considering, the Pontrjaginclasses are given by

p

(

SU(3)

U(1) × U(1)

)

= 1 , p

(

Sp(2)

S(U(2) × U(1))

)

= (1+x2)4 , p

(

G2

SU(3)

)

= 1 . (3.2)

9

SU(3)U(1)×U(1)

Sp(2)S(U(2)×U(1))

G2

SU(3)

G0 Z Z Z

G2 Z Z −G4

12Z 1

2Z −

G616Z 1

12Z 1

3Z

Table 2: Quantization condition on fluxes.

The first result follows from a theorem of Borel and Hirzebruch, according to which thePontrjagin class of a coset G/U , with U a maximal torus of G, is trivial. The latter twofollow from the identification of the two cosets topologically with CP3 and S6 respectively.The x that occurs is the generator of the integer cohomology of CP3. It follows that G6/2π

is quantized in multiples of 16

for the cosets SU(3)U(1)×U(1)

and G2

SU(3), and in multiples of 1

12for

Sp(2)S(U(2)×U(1))

. For G2

SU(3), we can go further. In [40], the following mod 2 relation among

Chern classes is derived

c3(E) = c1(E)c2(E) + Sq2c2(E) mod 2 .

Since G2

SU(3)has no 2- and 4-cohomology, it follows that c3(E) must be even for any vector

bundle on this space ([40] provide an index theory argument for this conclusion). Weconclude that on this coset, G6 is quantized in units of 1

3.

We turn to the second question raised above: given an element of H∗(X, Q) satisfying the

integrality conditions just discussed, when does it lie in the image of the map√

A ch(·),thus qualifying as a viable choice of flux? We will not provide a general answer, but addressthe following two subquestions which will be relevant in the next subsection.

Is it possible to have only G0 and G6 non-vanishing? It is a theorem (see e.g.Thm. V.3.25 in [41]) that the map (3.1) provides an isomorphism when the domain isextended to rational K-theory, K(X) ⊗ Q. It follows that any class in H∗(X, Q) lifts toa fractional K-theory class. Multiplying our choice of G0 and G6 with an appropriateinteger hence always provides a viable choice of flux.

Given G2 = 0, which G4 are permissible? When G2 vanishes, G4 is integrallyquantized. For the two cosets with non-trivial 2- and 4-cohomology, this is the onlyrestriction on G4, i.e. all of H4(X, Z) is a permissible choice for this flux. As pointedout in [40], this situation arises whenever the cohomology of the manifold is generated insecond degree. If we call the generators xi, line bundles Li exist with c1(Li) = xi. TheK-theory classes xij = Li ⊗ Lj − Li ⊕ Lj can then be used as building blocks for liftingany G4-flux, by

ch(xij) = xixj +1

2(x2

i xj + xix2j ) .

10

3.2 The solution

The N = 1 supersymmetry conditions for an AdS4 vacuum with internal SU(3) structurehave been determined by [42] (see [8, 43] for generalization to the SU(3)×SU(3) structurecontext). A nontrivial warp factor is not allowed, and the dilaton φ has to be constant.Furthermore, in our conventions6 the equations governing the H-field and the internal RRfield strengths read

H = (−1)s 2m

5eφ Re Ω , (3.3)

F0 = m , F2 = −f

9J + (−1)sie−φW2 , F4 =

3m

10J ∧ J , F6 =

f

6J ∧ J ∧ J ,

where the only nonvanishing purely imaginary torsion classes are W1 = (−1)s 4i9eφf and

W2. The only Bianchi identity which is not automatically satisfied is dF2 − HF0 = 0.This imposes

dW2 = ie2φ( 2

27f2 − 2

5m2)

Re Ω . (3.4)

The AdS cosmological constant is determined by

Λ = −3e2φ

(

m2

25+

f2

9

)

. (3.5)

Following work of [32], [26] showed that these equations can be solved on the cosets weintroduced in the previous section, by expanding all fields in forms invariant under theleft group action. We will repeat this analysis, but parametrize the solutions by thefluxes [G], as introduced in the previous subsection, rather than the parameter f and thedilaton. This is the favored approach as it allows us to take flux quantization into accountnaturally (from a 4d point of view, the distinction between fluxes and parameters suchas f and the dilaton is most striking, as the former correspond to charges, the latter toVEVs; in 10d, while fluxes can also be considered as VEVs, they are distinguished byencoding topological information).

We will focus on SU(3)U(1)×U(1)

for concreteness. This example is the most rich among thethree cosets we are considering, as it has the largest set of left-invariant forms, and thelargest cohomology.

The ansatz (2.14) already led to the expressions (2.15) for W1 and W2 in terms of themetric parameters va. It will prove convenient for this section to express the internalcomponent b of the B-field using the closed 2-forms (2.13),

b = b′1ω′1 + b′2ω′

2 + b′3ω3 .

6Our supergravity field strengths are as in [44]. We derive the susy conditions starting from an ansatzfor the two type IIA susy parameters ǫ1, ǫ2 wich assigns negative chirality to ǫ1 and positive chirality toǫ2. For the gamma matrices and the SU(3) structure we adopt the conventions listed in subsection A.2of [11]. The resulting equations (3.3) and the SU(3) torsion classes differ from the ones in [26] by just afew minus signs. The factor of (−1)s = ±1 arises in the following equations as unlike [42], we have fixedthe phase of Ω once and for all in (2.14); see also [43]. Both signs are consistent with supersymmetry.

11

Thus, b′1 and b′2 capture topological information about the B-field, while b′3 enters in H.Likewise, our ansatz for G is

G0 = m ,

G2 = m′1ω′1 + m′2ω′

2 ,

G4 = −e1ω1 − e2ω

2 − ξ dβ ,

G6 = −eω0 .

The equations of motion for G are complicated, and are encoded in the equations (3.3).By contrast, the Bianchi identities are already guaranteed by the ansatz (hence the useof primed forms).

To solve (3.3) in terms of the flux parameters, we begin by solving (3.4) in term of φ,invoking the relation between W1 and f ,

e2φ =5

16m2v1v2v3[6∑

a<b

vavb − 5∑

(va)2] .

For the rest of this section, φ will denote this solution.

Utilizing the equation for H, this allows us to solve for b′3 in terms of the metric param-eters,

b′3 = (−1)s+1 4m

5

√v1v2v3eφ

= (−1)s+1 m

|m|

√

√

√

√5

(

6∑

a<b

vavb − 5∑

(va)2

)

.

We next want to solve for f , starting with

F6 = G6 + B ∧ G4 +1

2B2 ∧ G2 +

1

3!B3 ∧ G0 =

f

6J ∧ J ∧ J . (3.6)

We eliminate the B3 term via

F4 = G4 + B ∧ G2 +1

2B ∧ B ∧ G0 =

3m

10J ∧ J

⇔ mB3 =3m

5B ∧ J ∧ J − 2B ∧ G4 − 2B2 ∧ G2 .

Hence,

f

6J ∧ J ∧ J = G6 +

2

3B ∧ G4 +

1

6B2 ∧ G2 +

m

10B ∧ J ∧ J ,

and substituting f into

F2 = G2 + B ∧ G0 = −f

9J + (−1)sie−φW−

2

12

yields three equations which can be solved for b′1, b′2 and ξ,

b′1 = (−1)s (5v1 − 3(v2 + v3))

√v1v2v3

4v2v3me−φ − m′1

m,

b′2 = (−1)s (5v2 − 3(v1 + v3))

√v1v2v3

4v1v3me−φ − m′2

m

We omit the expression for ξ, which is lengthy and not illuminating.

At this stage, we have expressed ξ, b′a, eφ in terms of va. Substituting these into the F4

equation,

G4 + G2 ∧ B +1

2B ∧ B ∧ G0 =

3m

10J ∧ J , (3.7)

yields three independent equations for va, two of which take the simple form

(v1 − v3)(v1v2 + v2v3 − 3v1v3)

v1v3− e2φ(

me1

I+ m′2(2m′1 + m′2)) = 0 ,

(v2 − v3)(v1v2 + v1v3 − 3v2v3)

v2v3− e2φ(

me2

I+ m′1(m′1 + 2m′2)) = 0 . (3.8)

The main new feature we wish to demonstrate, as compared to the Nearly Kahler analysisof [12], is the presence of several supersymmetric vacua of a given theory, i.e. upon a fixedchoice of fluxes. This phenomenon already occurs at ea = m′a = 0, which is a permissiblechoice of flux by the previous subsection. The third equation following from (3.7) heretakes the form

15eφ√

v1v2v3 e + (−1)s+18I v2v3(v2v3 − 3v1v2 − 3v1v3) = 0 .

It is easy to see that this system of equations has, aside from the Nearly Kahler solutionat7

v1 = v2 = v3 =

√15

2

(

1

20I

∣

∣

∣

e

m

∣

∣

∣

)1

3

,

the solution

v1 = v2 = 2v3 =

√15

4

(

1

2I

∣

∣

∣

e

m

∣

∣

∣

)1

3

,

as well as two others which arise upon cyclic permutation of v1, v2, v3.

The symmetry between the three metric parameters v1, v2, v3 can be broken by consideringbackgrounds with G4 flux. E.g., maintaining G2 = 0, we obtain from (3.8)

e1 6= 0 → v1 6= v3 ,

e2 6= 0 → v2 6= v3 ,

e1 6= e2 → v1 6= v2 .

We have checked numerically that e.g. at e1 6= 0, e2 = 0, solutions with v2 = v3 exist.

7Note that physicality (positivity of va) determines the appropriate choice of s depending on the signof e.

13

4 The dimensional reduction

4.1 The truncation scheme

As announced, we will adopt a reduction prescription in which the higher dimensionalsupergravity fields are expanded on a basis for the left-invariant tensors admitted by thecoset. This expansion basis was introduced in subsect. 2.1 for the three cosets (2.1).

We stress again that this G-invariant truncation does not coincide with a massless Kaluza-Klein ansatz. We can illustrate the differences between the two schemes e.g. by consider-ing the gauge vectors of the dimensionally reduced theory arising from the decompositionof the higher dimensional metric. The conventional massless Kaluza-Klein ansatz as-sociates a gauge vector of the truncated theory to each Killing vector on the compactmanifold, the gauge symmetry being inherited from the reparameterization invariance ofthe higher dimensional spacetime.8 On the other hand, the G-invariant ansatz preservesjust a subgroup of the full isometry group of the internal manifold G/H. The theory ofcompact left coset spaces endowed with a left-invariant metric (such are the cosets weconsider) states that in general the isometry group of G/H is G×N(H)/H, where N(H)is the normalizer of H in G, defined as N(H) := g ∈ G : gH = Hg . The G factor inG×N(H)/H is associated with the left action of G on the coset, while the N(H)/H factorderives from the right action of G. The Killing vectors generating the right isometriesare left-invariant, while this is not the case for the ones generating the left isometries.9 Itfollows that a left-invariant reduction ansatz keeps only the former, and the gauge groupdescending from the higher dimensional metric sector is just N(H)/H.

For the cosets we consider the G-invariant ansatz is particularly simple, because N(H)/Hturns out to be trivial. This can be seen either by observing that rank G = rank H [29],or by noticing that our cosets do not admit left-invariant vectors at all. We conclude thatno gauge vectors will descend from the dimensional reduction of the type II supergravityNSNS sector, and the whole (abelian) gauge group will be provided by the RR sector.This is analogous to what is realized in Calabi-Yau compactifications.

Though physically well motivated, dimensional reductions based on the full massless KKansatz have a drawback: they are generically inconsistent [46, 13]. Rare exceptions areknown, an example being the S7 reduction of [47] (see [48] for a discussion of consistentKK sphere reductions). The G-invariant reduction scheme is instead believed to provideconsistent truncations, due to the fact that the preserved invariant fields never generatethe truncated non-invariant modes. A further argument for consistency is that the substi-tution of a G-invariant ansatz guarantees the dropping of the dependence on the internalcoordinates y from the higher dimensional Lagrangian, see e.g. [49, 13] for more details.The consistency of the G-invariant scheme was explicitly shown in ref. [50] for a reduction

8In principle, nonvanishing background values of the non-metric supergravity fields may break thegauge symmetry to a subgroup of the isometry group, however this is guaranteed not to happen as faras these vevs are invariant under the isometries [13, pag. 16].

9A detailed discussion of the isometries of G/H can be found in section 2 of ref. [45].

14

of the pure gravity action. Recent related discussions can be found in [51] (for coset spacereductions of Einstein-Yang-Mills theories), in [52, 53] (for Scherk-Schwarz reductions ongroup manifolds), and in [54, 55] (for consistent reductions on spaces supporting AdS so-lutions, and their relation with the dual SCFT). However, an explicit check of consistencyin the context of SU(3) structure compactifications with fluxes had not been performedto date. In subsection 4.3 we will work out the reduction of the higher dimensional equa-tions of motion in detail, and prove the consistency of the truncation of the full type IIAbosonic sector for the cosets (2.1).

4.2 The 4d action

Following the reduction prescription for type IIA on SU(3) structure manifolds initiatedin [5], the complete 4d gauged N = 2 bosonic action has by now been derived [56, 57,58, 31, 14]. Here, we will use the notation of ref. [11]. Separating the contributions of theNSNS and RR sectors, the action S(4) arising from a reduction on our cosets (2.1) reads

S(4) = S(4)NS + S

(4)RR, with

S(4)NS =

∫

M4

( 1

2R4 ∗ 1 − 1

4e−4ϕdB ∧ ∗dB − dϕ ∧ ∗dϕ − Gabdta ∧ ∗dtb − VNS ∗ 1

)

, (4.1)

S(4)RR =

∫

M4

1

4ImNABF A ∧ ∗F B +

1

4ReNABF A ∧ F B − e2ϕ

4(Dξ ∧ ∗Dξ + dξ ∧ ∗dξ)

+1

4dB ∧

[

ξdξ − ξDξ + 2eAAA + ξ qaAa]

− 1

4mAeAB ∧ B − VRR ∗ 1

. (4.2)

The different quantities appearing in this 4d action are introduced in appendix A, wherewe also give some details about the derivation from the higher dimensional supergravity.The 4d degrees of freedom descending from the NSNS sector are the metric gµν , the 2–form B, the complex scalars ta = ba + iva and the 4d dilaton ϕ, defined in (A.2). TheRR sector yields the scalars ξ and ξ introduced in the first line of (A.13), as well as thegauge potentials AA, whose modified field strengths F A are defined in (A.14) (recall thatthe index A runs over (0, a) ).

The N = 2 action S(4) contains the gravitational multiplet (gµν , A0), a number of vector

multiplets (ta, Aa) (see table 1 for the coset dependent range of a), and one tensor multiplet(B, ϕ, ξ, ξ). When mA = 0 the antisymmetric tensor B becomes massless and can bedualized to a scalar, yielding the universal hypermultiplet. From Dξ = dξ − qaA

a itfollows that ξ is charged under the Aa, the charges being provided by the geometric fluxesqa given in table 1. The graviphoton A0 instead does not participate to this gauging (dueto the fact that the compactification manifolds (2.1) don’t allow for a flux of the NSNS3–form [5]).

The special Kahler metric Gab governing the kinetic terms for the scalars in the vectormultiplets is given in table 1, and further discussed in subsection A.1 of the appendix,

15

together with the period matrix NAB describing the kinetic and topological terms for thegauge potentials.

The full 4d scalar potential reads V = VNS + VRR. Reduction of the internal NSNS sectoron our coset spaces yields10

VNS ≡ −e2ϕ

2

(

R6 −1

2HyH

)

=e2ϕ

4Volqaqb

(

Gab − 3vavb + babb)

, (4.3)

where the 6d Ricci scalar R6 has been evaluated in terms of the torsion classes expressedin eq. (2.15) via the formula11 [59]

R6 =15

2|W1|2 −

1

2W2yW 2 , (4.4)

while for the internal NSNS 3–form we have H = d6b = baqaα.

The RR contribution to the scalar potential, obtained from the general expression givenin eq. (A.15) of the appendix, is

VRR = −e4ϕ

4

[

mAImNABmB+(eA+qAξ−mCReNCA)(ImN )−1 AB(eB+qB ξ−ReNBDmD)]

,

(4.5)where qA = (0, qa). Notice that while ξ appears in the potential, the other RR scalar ξ isa flat direction. Since the matrix ImN is negative, VRR is positive semi-definite.

4.3 Consistency of the truncation

We now prove the consistency of the dimensional reduction leading to the 4d action S(4)

introduced in the previous subsection. To this end, we plug the G-invariant reductionansatz into the bosonic equations of motion (EoM) of type IIA supergravity, and showthat these yield the EoM following from the reduced action S(4).

The reduction of the equations for the RR degrees of freedom was already described in thegeneral analysis of [11] and is summarized, for the specific compactification on the cosetspaces (2.1), in subsection A.2 of the appendix. In fact, the piece (4.2) of the 4d actionhas been established requiring its compatibility with the EoM for the 4d fields AA, ξ, ξ asobtained from the higher dimensional equations (A.10), (A.11). It follows that, as far theRR sector is concerned, the reduction is consistent by construction.

10For any pair of forms P,Q of degree k we define the contraction P yQ := 1k!Pm1...mk

Qm1...mk . Inour conventions for the Hodge ∗, we have

(

P yQ)

∗ 1 = P ∧ ∗Q. This also holds for the 10d spacetimeequations of the forthcoming subsection.

11An equivalent expression for R6 was given in [29] using a general formula relating the Riemann tensorof G/H to the structure constants of G. The 4 factor mismatch we have with respect to that expressionis due to the different normalization of the SU(3) structure constants already mentioned in footnote 3.

16

Hence, we just have to analyse the equations of motion for the NSNS degrees of free-dom, namely the B-field, the Einstein and the dilaton equations. For the democraticformulation of type IIA supergravity [44] in string frame, these read

d(e−2φ ∗ H) − 1

2[F ∧ ∗F]8 = 0 , (4.6)

RMN + 2∇M∂Nφ − 1

2ιM HyιNH − e2φ

4

10∑

k=0

ιM F(k)yιN F(k) = 0 , (4.7)

R − 1

2HyH + 4

(

∇2φ − ∂Mφ∂Mφ)

= 0 , (4.8)

where the hat denotes 10d quantities, F ≡∑10k=0 F(k) is the sum of the RR field-strengths,

and M, N are 10d spacetime indices.

B-field EoM

The B-field EoM (4.6) is an 8–form equation. Its expansion in the left-invariant forms onM6 yields two independent equations: the first exhibiting two indices along 4d spacetimeM4 and 6 indices along M6, and the second with 4 indices along M4 and 4 indices alongM6. We get no equation with 5 indices along M6 due to the absence of invariant 5-formson the cosets (2.1). Concretely, recalling (A.11) we rewrite the RR piece of (4.6) as

[F ∧ ∗F]8 = [F ∧ λ(F)]8 = [G ∧ λ(G)]8 .

Expanding B as in (A.4) and G as in (A.12), we see that eq. (4.6) reduces to

[

d(e−4ϕ ∗ dB) + GA(0)G(2)A − G(0)AGA

(2) + G(1) ∧ G(1)

]

ω0 = 0 (4.9)

and

− 4d4(Gab ∗4 d4bb)ωa + e−2φ+4ϕvol4 ∧ d6(∗6d6b) + (4.10)

+[

G0(0)G(4)a + G0

(4)G(0)a −KabcGb(0)G

c(4) − G0

(2) ∧ G(2)a +1

2KabcG

b(2) ∧ Gc

(2)

]

ωa = 0 ,

where the 4d forms G(p), G(p) are expressed in (A.13), and we used ωa ∧ ωb = −Kabcωc,

with the Kabc given in (A.7).

Eq. (4.9) provides the EoM for the 2–form B in 4d. It already appeared in section 5 ofref. [11], where it was employed in order to deduce the 4d action S(4) written in subsection4.2 above. It follows that, on the same footing as the RR equations, consistency of thisequation with the action S(4) is guaranteed by construction.

Eq. (4.10) (which was not analysed in [11]) corresponds to the EoM for the 4d scalars ba

defined by the expansion of the internal B-field b on the basis 2–forms. Using d6 ∗6 d6b =

17

qbbbqaω

a, substituting the expressions (A.13) for G(2), G(2), G(4), G(4) and the definition(A.14) of F A, eq. (4.10) reads

4∇µ(Gab∂µbb) − e2ϕ qbb

bqa

Vol− ImNaB ∗ (F 0 ∧ ∗F B) − ReNaB ∗ (F 0 ∧ F B)

+1

2Kabc ∗ (F b ∧ F c) + e4ϕ

[

G0(0)(ImNG(0) − ReNL)a − G(0)aL

0 + KabcGb(0)L

c]

= 0,

where we denote L ≡ (ImN )−1(G(0) −ReNG(0)). Recalling the form of ImN and ReNin (A.8) and (A.9), as well as VNS in (4.3) and VRR in (A.15), one checks that this equationcan be reformulated as

2∇µ(Gab∂µbb)− 1

4∂baImNBC ∗ (F B∧∗F C)− 1

4∂baReNBC ∗ (F B∧F C)−∂ba(VNS +VRR) = 0

which is precisely the EoM obtained varying S(4) in (4.1), (4.2) with respect to ba.

10d Einstein equation

We first deal with the term RMN + 2∇M∂Nφ in eq. (4.7). Starting from the G–invariantmetric ansatz (A.1) and recalling that the 4d dilaton ϕ(x) satisfies (A.3), we derive thefollowing decomposition12

Rµν + 2∇µ∂νφ = Rµν −1

4gmpgnq∂µgmn∂νgpq − 2∂µϕ∂νϕ − gµν∇2

4 ϕ ,

Rµn = 0 = ∇µ∂nφ ,

Rmn + 2∇m∂nφ = Rmn +1

2e−2ϕ

(

gpq∂µgmp∂µgnq −∇2

4 gmn

)

. (4.11)

Taking the trace, we get

R + 4∇2φ− 4∂Mφ∂Mφ = e−2ϕ(

R4 + e2ϕR6 −1

4gmpgnq∂µgmn∂

µgpq − 2∇24 ϕ− 2∂µϕ∂µϕ

)

.

(4.12)In the previous expressions, quantities labeled with 4 or 6 are associated to (M4, gµν) or(M6, gmn) respectively. The 4d indices on the r.h.s. are raised using the rescaled metric gµν

of eq. (A.1). Notice that all the terms depend just on xµ: indeed, thanks to G-invariance,the whole dependence on the internal coordinates drops out.

Let’s now consider the µν components of the 10d Einstein equation (4.7). Using (4.11),(4.12) we find (we reinstate in the Einstein equation the term proportional to gµν , which

12The nonvanishing higher dimensional Christoffel symbols are:

Γρµν = Γρ

µν + ∂µϕδρν + ∂νϕδ

ρµ − gµν∂

ρϕ , Γρmn = −1

2e−2ϕ∂ρgmn , Γp

µn =1

2gpq∂µgnq , Γp

mn = Γpmn .

In the derivation of Rµn = 0 we assume ∇mepn = 0.

18

actually vanishes thanks to the dilaton EoM (4.8) ),

Rµν + 2∇µ∂νφ − 1

2ιµHyινH − 1

2gµν

(

R + 4∇2φ − 4∂ρφ∂ρφ − 1

2H2)

=

= Rµν −1

4e−4ϕHµρσH ρσ

ν − 2∂µϕ∂νϕ − 2Gab∂(µta∂ν)t

b (4.13)

− gµν

( 1

2R4 −

1

24e−4ϕHµνρH

µνρ − ∂µϕ∂µϕ − Gab∂µta∂µtb − VNS

)

.

For the RR piece, taking into account all the terms of the expansion described in subsec-tion A.2 of the appendix, we arrive at

−e2φ

4

10∑

k=0

ιµF(k)yινF(k) =1

2ImNABιµF

AyινF

B − 1

2e2ϕ(DµξDνξ + ∂µξ∂ν ξ)

−gµν

1

4ImNABF A

yF B − e2ϕ

4[(Dµξ)

2 + (∂µξ )2] − VRR

. (4.14)

From (4.13), (4.14) we see that the equation arising from the µν components of (4.7)precisely reproduces the 4d Einstein equation following from S(4).

Since there are no left-invariant 1–forms on the cosets (2.1), the 10d Einstein equationwith µn indices is trivialized by our left-invariant truncation prescription, and does notyield any constraint at the 4d level. Indeed, one can check that all the µn terms in (4.7)vanish once the truncation ansatz is plugged in.

Finally, we study the purely internal components of (4.7) in flat mn indices. Dependingon which of the cosets (2.1) we consider, these yield just one, two or three 4d scalarequations, labeled by the index a. On our cosets, any left-invariant symmetric rank-2tensor has the same diagonal structure as the invariant metric gmn given in subsection2.1. Furthermore, the left-invariant Ricci tensor on coset spaces satisfies Rmn = ∂

∂gmn R6.

Focusing for definiteness on SU(3)U(1)×U(1)

, we have (recall Gab in table 1)

R2a−1 2a−1 ≡ R2a2a = −1

8Gab∂vbR6 , a = 1, 2, 3 .

Then, using the last line of (4.11), we get

R2a2a + 2∇2a∂2aφ− 1

2ι2aHyι2aH =

e−2ϕGab

4

[

− 2∇µ(Gbc∂µvc) + ∂vbGcd∂µt

c∂µtd + ∂vbVNS

]

.

(4.15)Concerning the RR term, a tedious computation gives

−e2φ

4

10∑

k=0

ι2aF(k)yι2aF(k) =e−2ϕGab

4

[

∂vbVRR − 1

4∂vb(ImNCD)F C

yF D]

. (4.16)

Analogous steps can be repeated for the cosets Sp(2)S(U(2)×U(1))

and G2

SU(3), leading to the same

r.h.s. of the equations here above.

19

From (4.15), (4.16) we conclude that the components of the 10d Einstein equation (4.7)with two internal indices precisely match the EoM for the scalars va following from S(4):

− 2∇µ(Gab∂µvb) + ∂vaGbc∂µt

b∂µtc + ∂va(VNS + VRR) − 1

4∂va(ImNBC)F B

yF C = 0 .

Dilaton equation

Subtracting the trace over the µν components of (4.7) from the 10d dilaton equation (4.8),we eventually obtain

2∇24ϕ +

1

6e−4ϕHµνρH

µνρ − e2ϕ

2

[

(Dµξ)2 + (∂µξ)2]

− 2VNS − 4VRR = 0 , (4.17)

which is the EoM for the 4d dilaton ϕ following from S(4).

This concludes the consistency proof of the dimensional reduction.

5 The 4d potential via N = 2

In this section, we recast the scalar potential obtained in (4.3) and (4.5) in 4d N = 2language. In this framework, given the prepotentialF governing the special geometry dataof the vector multiplet sector and the quaternionic metric huv of the hypermultiplet sector,the potential is uniquely determined by the gauged isometries of huv. This structure allowsus to incorporate string loops into our considerations, which correct the hypermultipletmetric. As the 4 dimensional quaternionic metrics with the isometry structure imposedby our compactifications are highly constrained, we use the results of [65, 67] to writedown the general form of the all-loop string corrected potential in subsection 5.2. Weanalyse this potential further in subsection 6.3.

The general form of the potential in 4d N = 2 gauged supergravity is [10, 60, 61, 62]

V = 4eKhuv(XAku

A − kuAFA)(XBkuB − kuBFB)

−[

1

2(ImN )−1 AB + 4eKXAXB

]

(PxA − PCxNCA)(Px

B − PDxNDB) . (5.1)

The coordinates X, the prepotential F , and the gauge coupling matrix N encode specialgeometry data and are discussed further in appendix A. huv refers to the universal hy-permultiplet metric, which is expressed in terms of the quaternionic vielbein componentsas

h = u ⊗ u + v ⊗ v .

We will denote the quaternionic coordinates collectively by qu. kuA and kuA are the com-

ponents of the Killing vectors describing the isometries of the hypermultiplet metric beinggauged by the Ath gauge vector. The Sp(1) factor ω of the spin connection of the hyper-multiplet metric enters in the potential via its relation to the Killing prepotentials. For

20

the case that the 3 components of the curvature of ω each are invariant under an isometryku∂qu of the metric, the corresponding Killing prepotential is given by [60, 63]

Px = ωxuku . (5.2)

In this case, one can rewrite the potential in a more convenient form. Introducing

QuA = ku

A − kuBNBA ,

we obtain

V = QuAQv

B

[

4eKXAXB(

u⊗ u+v⊗ v)

uv−(

4eKXAXB +1

2(ImN )−1 AB

)

∑

x

(

ωx ⊗ωx)

uv

]

.

(5.3)

5.1 Tree level

At tree level, the quaternionic vielbein is given by [64]13

u =1

2eϕ(dξ − idξ) ,

v = dϕ − ie2ϕ

2

(

da + ξdξ)

.

The Sp(1) connection has the following form in terms of these quaternionic vielbein com-ponents14

ω1 = i(u− u) , ω2 = −(u + u) , ω3 =i

2(v − v) . (5.4)

In the class of theories we are considering, the isometries being gauged are described bythe following Killing vectors

kA =√

2

(

eA∂

∂a+ qA

∂

∂ξ

)

,

kA =√

2mA ∂

∂a. (5.5)

Since Qu does not contain a non-vanishing entry for u = ϕ, the real part of v does notenter upon contraction with Qu, hence we can substitute

∑

x

(

ωx ⊗ ωx)

∼ 4u ⊗ u + v ⊗ v

13ϕ, ξ, ξ were introduced above. The coordinate a is related to the dual aB of the spacetime component

of the B-field via aB = a+ ξξ2 .

14The components ωx of the Sp(1) curvature ω should not be confused with the expansion forms ωa.

21

in the potential, obtaining

V = QuAQv

B

[

− e2ϕ(1

2(ImN )−1 AB + 3eKXAXB

)(

dξ2 + dξ2)

uv

−1

8e4ϕ(ImN )−1 AB

(

da + ξdξ)2

uv

]

.

This coincides with (4.3) and (4.5) obtained above via reduction from 10 dimensions.

5.2 All string loop

For the case of the universal hypermultiplet with 3 isometries, the quaternionic metric is ofthe Calderbank-Pedersen form [65]. It comes in a 1-parameter family [66, 67], determinedby

u =

√

ρ2 + c

2(ρ2 − c)(dξ − idξ) ,

v =ρ

2(ρ2 − c)√

ρ2 + c

[

2ρ2 + c

ρdρ + i(da + ξdξ)

]

. (5.6)

The metric at string tree level lies at c = 0, and the variable identification

ρ = e−ϕ

takes up back to the expression for the metric introduced above.15

In terms of the quaternionic vielbein components (5.6), the Sp(1) connection of theCalderbank-Pedersen metric is [65]

ω1 =ρ

√

ρ2 + ci(u− u) = − ρ

ρ2 − cdξ ,

ω2 = − ρ√

ρ2 + c(u + u) = − ρ

ρ2 − cdξ ,

ω3 =

√

ρ2 + c

ρ

i

2(v − v) = − 1

2(ρ2 − c)(da + ξdξ) . (5.7)

The N = 2 potential (5.3) for this choice of metric becomes

V =Qu

AQvB

(ρ2 − c)2

[

(

− 1

2(ImN )−1 AB − 3eKXAXB

)

ρ2(dξ2 + dξ2)uv

−1

8(ImN )−1 AB(da + ξdξ)2

uv + c eKXAXB(dξ2 + dξ2)uv

− c

ρ2 + ceKXAXB(da + ξdξ)2

uv

]

. (5.8)

15The coordinates used in [67] are related to our choice via ψ = a+ξξ2

, η = − ξ2, φ = ξ.

22

In the case of Calabi-Yau compactifications, the metric is corrected away from c = 0 inpassing from tree level to one-loop [67]. Beyond 1-loop, all corrections can be capturedby field redefinitions. This means that the quaternionic metric (i.e. the value of c)remains unchanged, the identification ρ = e−ϕ however is modified (note that the isometrystructure of the metric determines the identification of the other 3 Calderbank-Pedersencoordinates with the 10d variables as indicated in footnote 15; this is why we have notintroduced separate notation for them).

To study perturbative string corrections in the case of interest, let us review the argu-ment of [67]. The 1-loop correction to the four dimensional Einstein-Hilbert term can bedetermined by reduction of the 1-loop R4 correction in 10d.16 In the normalization of[67], this yields

SEinstein−Hilbert =

∫

d4x√

g(

e−2φ − 4ζ(2)χ

(2π)3

)

R .

Unfortunately, the full 1-loop corrected 10d action is not available as a means towardsobtaining the 1-loop completion of the 4d action. Nonetheless, after parametrizing theignorance regarding this action and comparing to the 4d effective action obtained bychoosing the Calderbank-Pedersen metric on the universal hypermultiplet scalar manifold,[67] finds that only two possible values for c are possible,

c = 0 or c = −4ζ(2)χ

(2π)3, (5.9)

with χ the Euler characteristic of the Calabi-Yau. A perturbative string calculationthen establishes that it is the latter value that is correct beyond tree level. Such acalculation in the case of the coset backgrounds with RR-flux that we are interested inis very challenging, and beyond the scope of this work. However, the first part of theanalysis of [67] goes through also for these more general backgrounds. In particular, the10d R4 term is proportional to [67]

t8t8R4 +

1

4E8 .

The first term is shorthand for t8t8R4 = tM1···M8tN1···N8RM1M2N1N2

· · ·RM7M8N7N8, which

is expanded in terms of scalars built out of contractions of four Riemann tensors in eq.(A.12) of [67]. The second term can be written compactly in form notation as

E8 ∼ ΩAB ∧ ΩCD ∧ ΩEF ∧ ΩGH ∧ ∗(eA ∧ · · · ∧ eH) ,

with ΩAB = 1

2RA

BCDeCeD the curvature 2-form and eA, A = 1, . . . , 10 a local coframebasis. From the expansion of the t8 term in [67], we see that in each scalar invariant,

16As with all such arguments, we are relying on the off-shell continuation of an on-shell string compu-tation. It would be desireable to back this line of reasoning up with an explicit string computation onthe background in question. We thank Pierre Vanhove for discussions on this point.

23

contractions pair at least two Riemann tensors. Hence, this term does not contribute tothe 4d Einstein-Hilbert term upon reduction. The contribution from E8 to the Einstein-Hilbert term stems, exactly as in the Ricci flat case, from

Ωab ∧ ∗4(ea ∧ eb) ∧ Ωmn ∧ Ωpq ∧ Ωrs ∧ ∗6(e

m ∧ · · · ∧ es) ,

with a, b flat spacetime and m, n, . . . flat internal indices. We recognize the internal contri-bution as proportional to the 6 dimensional Euler density. The conclusion of our analysisis hence that in generalizing beyond Calabi-Yau manifolds, the same two possibilitiesfor the Calderbank-Pedersen parameter c exist as in the Calabi-Yau case (and await aperturbative string calculation as arbiter).

6 Non-supersymmetric vacua

As an application of our consistent truncation result, we will search for non-supersymmetricvacua of the 4d effective action. By the analysis of section 4, these are guaranteed to liftto 10d solutions.

6.1 Tree level

The potential we obtained at tree level above has the form

V = A1e2ϕ + A2e

4ϕ , (6.1)

with

A1 = −QuAQv

B

(1

2(ImN )−1 AB + 3eKXAXB

)(

dξ2 + dξ2)

uv,

A2 = −QuAQv

B

1

8(ImN )−1 AB

(

da + ξdξ)2

uv. (6.2)

Minimizing the potential with regard to the 4d dilaton yields [68]

Vϕ = − A21

4A2.

As A2 is positive definite, the potential at tree level is negative semi-definite on-shell.In fact, this result generalizes immediately to any hypermultiplet metric of the generalform [64] that arises upon Calabi-Yau and SU(3) structure compactifications, and therespective gaugings. The corresponding potential is obtained by appropriately modifyingu and v in (6.2). A2 hence remains positive also in this more general case.

We have thus proved that N = 2 gauged supergravity as it arises in Calabi-Yau likecompactifications at string tree level (i.e. with hypermultiplet metric as given in [64], andgaugings of axionic isometries) does not permit de Sitter solutions. Due to the consistency

24

of the truncation, this 4d result also follows from the 10d no-go theorem of Maldacena-Nunez [69]. Note however that our 4d reasoning continues to hold for an arbitrary vectormultiplet sector, i.e. including all possible worldsheet instanton corrections.

The two contributions to (6.1) arise upon compactification from the NSNS and the RRsector respectively, see (4.3) and (4.5). The positivity of A2 is also manifest here.

6.2 Non-supersymmetric Nearly Kahler companions

The 10d analysis of subsection 3.2 reveals that, given a choice of the RR fluxes G0 and G6,with all the other fluxes vanishing, there exists a single Nearly Kahler supersymmetricvacuum on the cosets (2.1). This solution is also recovered adopting the 4d approach, asdiscussed in [31, 12].

It is possible to show that, under the same conditions, the 4d tree level scalar potentialV also admits non-supersymmetric Nearly Kahler extrema. In the following formulae, weintroduce the sum of the geometric fluxes q ≡∑a qa, we rename the RR fluxes as e0 → e ,m0 → m, and we call the equal va and the equal ba respectively v and b.

We obtain three Nearly Kahler extrema, lying at

v =

√15

2

(

1

20I

∣

∣

∣

e

m

∣

∣

∣

)1/3

, b =1

2

(

1

20I

e

m

)1/3

, ξ =24Imb2

q, e2ϕ =

5q2

48I2m2v4,

(6.3)

v =√

3

(

1

20I

∣

∣

∣

e

m

∣

∣

∣

)1/3

, b = −(

1

20I

e

m

)1/3

, ξ = −12Imb2

q, e2ϕ =

q2

12I2m2v4,

(6.4)and

v =

(

1√5I

∣

∣

∣

e

m

∣

∣

∣

)1/3

, b = 0 = ξ , e2ϕ =5q2

36I2m2v4. (6.5)

By comparing to section 3.2, we learn that the only extremum preserving supersymmetryis (6.3).

Thanks to the consistency of the reduction, the non-supersymmetric extrema of V foundhere also solve the 10d equations of motion, and actually turn out to coincide with the so-lutions previously found in ref. [70] via a 10d approach (see subsection 11.4 therein).

Unlike the situation for the supersymmetric solution (6.3), for (6.4) and (6.5) stability isof course no longer guaranteed. As in any truncation scheme, a full stability analysis canonly take place in the higher dimensional theory. What we can offer in our 4 dimensionaltheory is a stability analysis with regard to the modes we retain. To this end, we rescalethe scalar fields17 (va, ba, ϕ, ξ) to obtain canonically normalized kinetic terms, and thendiagonalize the mass matrix at the respective solutions.

17Note that the shift symmetry of a and ξ is gauged, the background value of these fields is hence agauge choice.

25

Figure 1: The potential for G2

SU(3): we plot the rescaled potential e

5

3m1

3 I4

3V as a function of (Im/e)1

3 b

and |Im/e| 13 v, at the extremum of ϕ and ξ. The deepest minimum corresponds to solution (6.4). Thecut of the plot at V = 0 is due to the constraint eϕ(b,v) > 0.

The case G2

SU(3)is depicted in figure 1: the first two extrema (6.3) and (6.4) are minima,

while the remaining extremum is a saddle point. For SU(3)U(1)×U(1)

and Sp(2)S(U(2)×U(1))

, (6.4) isa minimum, whereas due to modes leading away from the Nearly Kahler locus va = vfor all a, (6.3) is merely a saddle point, as is (6.5). To analyse stability, we compare themagnitude of the negative masses at the saddle points with the Breitenlohner-Freedmanbound

m2tachyonic ≥ −3

4|V | .

All extrema (including the saddle point depicted in figure 1) prove stable.

Finally, we remark that α′ and string loop corrections can be safely neglected for thesolutions above by tuning the RR fluxes e and m in such a way that the internalvolume Vol ≡ v3I ∼ e/m becomes sufficiently large and the string coupling constant

eφ ≡ eϕ√

Vol ∼ e−1

6 m− 5

6 becomes small (recall the definition (A.2) of the 4d dilaton). Wecan study moderately large string coupling by invoking the corrected potential (5.8). Anumerical analysis indicates that all three AdS extrema survive string loop corrections.For the supersymmetric extremum, we push beyond numerics in appendix B, and establishanalytically that it persists, as expected, in the face of string loop corrections.

6.3 de Sitter vacua at all string loop order?

In face of the no-go result for de Sitter vacua obtained in subsection 6.1, we would like toanalyse how loop corrections modify the outcome of this study. Of course, to guaranteethe consistency of the truncation, the analysis in section 4 must be extended beyondthe two derivative case. However, the arguments put forth in subsection 4.1 in favor ofconsistency apply to the additional terms as well. We will also assume in this sectionthat c 6= 0, as in the Calabi-Yau case. Note that by the results above, we can performan (almost) complete analysis of the full loop corrected potential. The identification of

26

the physical coordinate ϕ and the Calderbank-Pedersen coordinate ρ, which is modifiedorder by order in the string coupling and is not available, merely enters in identifying therange of the CP coordinate, see below. Away from very strong coupling (in which braneinstanton corrections would have to be considered regardless), this does not affect thesearch for de Sitter minima.

Focusing on the ρ dependence of the potential (5.8) and taking the obvious positivityconstraints on the coefficients into account does not rule out de Sitter vacua. One canthen proceed to derive various constraints on these coefficients. E.g., by noting that thepotential (5.8) has the form

V (ρ) = P (ρ)Q(ρ) ,

with P (ρ) = 1(ρ2−c)2

, we obtain

V (ρ0) = −P 2

P ′Q′|ρ0

=Qu

AQvB

2(ρ20 − c)

[

(

− 1

2(ImN )−1 AB − 3eKXAXB

)

(4dξ2)uv

+c

(ρ20 + c)2

eKXAXB(da + ξdξ)2uv

]

,

where ρ0 signifies the value of ρ at a minimum of the potential. Since c is negative for thecosets we are considering, a de Sitter vacuum requires the first term in the square bracketto be positive at the minimum of the potential. This term is proportional to the tree levelNSNS contribution to V , given in eq. (4.3). Hence, our necessary condition translatesinto the following inequality involving the internal NSNS 3–form and Ricci scalar

HyH − 2R6 > 0 .

Recalling eq. (4.4), this is obviously true whenever the non-vanishing SU(3) torsionclasses satisfy 15|W1|2 < W2yW 2. For the simple case of Nearly Kahler manifolds (i.e.when W2 = 0) the inequality is however non-trivial, and reads 3b2 − 5v2 > 0.

We hope to return to a more complete analysis of the all loop corrected potential in thenear future.

Acknowledgements

We would like to thank Alessandro Tomasiello for collaboration in the initial stages ofthis project. We also acknowledge useful discussions with Adel Bilal, Paul Koerber, LucaMartucci, Ruben Minasian, Dimitrios Tsimpis and Pierre Vanhove. DC gratefully thanksthe Service de Physique Theorique et Mathematique de l’Universite Libre de Bruxelles,where part of this work was done, for hospitality and financial support. DC and AKthank the Erwin Schrodinger Institute in Vienna for hospitality during the “Mathematical

27

Challenges in String Phenomenology” workshop. DC and AK are supported in part by theEU grant MRTN-CT-2004-005104. In addition, DC is partially supported by the EU grantMRTN-CT-2004-512194, by the French grant ANR(CNRS-USAR) no.05-BLAN-0079-01and by the “Programme Vinci 2006 de l’Universite Franco-Italienne”. AK is supportedin part by l’Agence Nationale de la Recherche under the grants ANR-06-BLAN-3 137168and ANR-05-BLAN-0029-01.

A Details of the dimensional reduction

The G-invariant reduction ansatz strongly constrains the dependence of all the higherdimensional fields on the G/H coordinates, relegating it into the coframe em introducedin subsection 2.1. In particular, the most general G-invariant 10d metric is (here and inthe following, the hat denotes 10d fields):

ds2 = e2ϕ(x)gµν(x)dxµ ⊗ dxν + gmn(x)em(y) ⊗ en(y) , (A.1)

where xµ and ym are respectively coordinates on the 4d spacetime and the internal man-ifold M6, and gmn satisfies the G-invariance condition discussed in subsection 2.1. Com-ponents of the 10d metric with mixed 4d-6d indices are not allowed since there are noleft-invariant 1–forms on our coset manifolds (2.1). Since the invariant scalars on thecoset are necessarily constant, a nontrivial warp factor is also not permitted (see [71, 72]for recent discussions of a non-trivial warp factor in the N = 1 context). The Weyl factore2ϕ(x) in front of the 4d metric is needed in order to obtain a canonical lower dimen-sional Einstein-Hilbert term

∫

M4

vol4R4 from the string frame higher dimensional action∫

M10

vol10e−2φR, with

ϕ(x) = φ(x)− 1

2log

∫

M6

d6y√

g6 , (A.2)

where φ(x) is the 10d dilaton and√

g6 ≡√

det gmn(x, y) =√

det gmn(x) | det epq(y)| .

Notice that, thanks to this factorization of the x and y dependence, ∂µ log√

g6 does notdepend on the internal coordinates, and

∂µϕ = ∂µφ − 1

2∂µ log

√g6 . (A.3)

The ansatz for the 10d supergravity field strengths must be chosen consistently with theirBianchi identities. For instance, from the Bianchi identity dF2 = HF0, one sees that ifF0 6= 0, then the NSNS 3–form H has to be exact: H = dB, with a globally defined2–form potential B. The most general B respecting left-invariance on M6 is

B = B + b , (A.4)

where B(x) is along 4d spacetime, while b(x, y) = ba(x)ωa(y) lives on M6 (the left-invariant2–forms ωa are given in subsection 2.1).

We deal with the expansion of the RR fields in subsection A.2.

28

A.1 Special Kahler geometry from the NSNS sector

Combining the 2–form J of subsection 2.2 and the internal NS field b we introduce t =b + iJ , whose expansion t = taωa on the basis 2–forms defines the complex 4d scalarsta = ba + iva. The associated kinetic term is determined by

1

8gmpgnq

(

∂µgmn∂µgpq + ∂µbmn∂µbpq

)

=1

4Vol

∫

M6

∂µt ∧ ∗∂µt = Gab∂µta∂µtb , (A.5)

where the l.h.s. originates from the reduction of the 10d Ricci scalar and H2 terms,while the σ-model metric Gab was introduced in eq. (2.12). The first equality in (A.5)is derived recalling that the internal metric is fixed by the forms J and Ω defining theSU(3) structure: indeed, calling I the almost complex structure induced by Ω, we havegmn = JmpIp

n. Notice that we get no contribution from the variation of I since theassociated Ω, given in eq. (2.14), is rigid.

The metric Gab is special Kahler: indeed, it can be obtained via Gab = ∂2K∂ta∂tb

from theKahler potential

K = − log4

3

∫

J ∧ J ∧ J = − log 8Vol . (A.6)

It in turn is determined by a prepotential F via the special Kahler geometry formula

K = − log i(XAFA − XAFA ), where XA ≡ (X0, Xa) = (1,−ta) and FA = ∂F(X)

∂XA .

For each of the cosets we consider, the explicit expressions of Gab and Vol are given intable 1. The (cubic) prepotential reads

F(X) =1

6Kabc

XaXbXc

X0,

where the non-vanishing triple intersection numbers Kabc :=∫

ωa ∧ ωb ∧ ωc (recall the2–forms ωa in subsection 2.1) are

K123 = I for SU(3)U(1)×U(1)

K112 = 2I for Sp(2)S(U(2)×U(1))

K111 = 6I for G2

SU(3).

(A.7)

The period matrixNAB of special Kahler geometry is given by the formula (see e.g.[73])

NAB = FAB + 2iIm (FAC)XCIm (FBD)XD

XEIm (FEF )XF, where FAB ≡ ∂2F

∂XA∂XB.

Equivalently, we can directly obtain it from the coset geometry via [20]:

(ImN )−1 AB = −∫

〈ωA, ∗bωB〉 , [ReN (ImN )−1] B

A = −∫

〈ωA, ∗bωB〉 ,

[ImN + ReN (ImN )−1ReN ]AB = −∫

〈ωA, ∗bωB〉 ,

29

with ∗b( · ) ≡ e−b ∗λ(eb · ) . The operator λ and the pairing 〈 , 〉 were defined below (2.10).

We obtain the matrices

ImN = −Vol

(

1 + 4Gabbabb 4Gabb

b

4Gabbb 4Gab

)

, (A.8)

ReN = −(

13Kabcb

abbbc 12Kabcb

bbc

12Kabcb

bbc Kabcbc

)

. (A.9)

A.2 The RR sector

In order to reduce the RR sector we specialize the general procedure described in section5of ref. [11] for M6 corresponding to our coset spaces. Adopting the democratic formulationof type IIA supergravity [44], the RR degrees of freedom can be encoded in a field strengthG consisting of a formal sum of forms of all possible even degrees, satisfying

Bianchi identity : dG = 0 (A.10)

self-duality constraint : F = λ(∗F) , where F ≡ eBG and λ(F(k)) = (−)k

2 F(k). (A.11)

Due to the self-duality constraint, the equations of motion for the RR degrees of freedomare equivalent to the Bianchi identities.

We implement the reduction ansatz by expanding G on the basis of left-invariant internalforms introduced in subsection 2.1,

G = (GA(0) +GA

(2) +GA(4))ωA − (G(0)A + G(2)A + G(4)A)ωA + (G(1) + G(3))α− (G(1) + G(3))β.

(A.12)G(p)(x) and G(p)(x) are p–forms in 4d spacetime. Plugging this expansion into eqs.(A.10),(A.11), and going through the derivation of [11], one identifies the 4d variables

GA(0) = mA , G(0)A = eA + qA ξ (A.13)

G(1) = Dξ ≡ dξ − qaAa , G(1) = dξ

GA(2) = dAA , G(2)A + BG(0)A = ImNAB ∗ F B + ReNABF B

G(3) = −B ∧ Dξ + e2ϕ ∗ dξ , G(3) = −B ∧ dξ − e2ϕ ∗ Dξ

GA(4) + B ∧ GA

(2) +1

2B2 GA

(0) = e4ϕ[

(ImN )−1(G(0) − ReNG(0))]A ∗ 1

G(4)A + B∧G(2)A +1

2B2G(0)A = e4ϕ

[

− ImNG(0) + ReN (ImN )−1(G(0) − ReNG(0))]

A∗ 1

where the propagating fields are the two real scalars ξ, ξ and the 1–forms AA. We alsointroduced the modified field strengths

F A ≡ dAA + mAB . (A.14)

30

Furthermore we introduce qA = (0, qa), the qa being the geometric fluxes defined in sub-section 2.1.4, while mA, eA are constant flux parameters satisfying qam

a = 0. Notice thatone of the ea is redundant, since it can be eliminated via a constant shift of ξ . Thisreflects the fact that on our cosets the linear combination qaω

a is exact (see eq. (2.11)),and therefore doesn’t support any flux.

The residual content of (A.10)–(A.12) not included in eqs. (A.13) consists of a set ofequations to be read as the EoM for ξ, ξ and AA. We use these equations to reconstructthe 4d action S

(4)RR of subsection 4.2. In particular, we infer the RR contribution to the 4d

scalar potential,

VRR = −e4ϕ

4

[

G(0)ImNG(0) + (G(0) − G(0)ReN )(ImN )−1(G(0) − ReNG(0))]

. (A.15)

Substitution of the explicit expressions for G(0) and G(0) given in (A.13) yields eq. (4.5).

As a last remark, we stress that the whole procedure of section5 of [11] applies here with noneed to take any integral over M6. In other words, once the left-invariant truncation ansatzhas been plugged in, the dependence of eqs. (A.10), (A.11) on the internal coordinatesautomatically factorizes out.

B String loop corrections to the N = 1 vacua

In this appendix, we study how string loop corrections affect the tree level supersymmetricAdS4 solutions of type IIA supergravity compactified on the cosets (2.1).

The N = 1 equations arise by requiring the vanishing of the fermionic (i.e. gravitino-,hyperino- and gaugino-) variations under a single linear combination of the two N = 2susy parameters. These conditions have been spelled out in [20] for general SU(3)×SU(3)structure compactifications, and solved in [12] for the subclass of Nearly Kahler mani-folds. Here, we extend the latter analysis employing the string loop corrected quaternionicvielbein (5.6) and the associated Sp(1) connection. In particular, the Killing prepoten-tials associated with our electric and magnetic gaugings of the quaternionic isometriesbecome, recalling relation (5.2), the Killing vectors (5.5), and the Calderbank-PedersenSp(1) connection (5.7),

P1A = −

√2ρ

ρ2 − cqA , P1A = P2

A = P2A = 0 ,

P3A = −

√2

2(ρ2 − c)(eA + ξqA) , P3A = −

√2

2(ρ2 − c)mA . (B.1)

The tree level Killing prepotentials are recovered by taking c = 0 (recall the possiblevalues of c, given in (5.9)), together with the identification ρ2 = e−2ϕ. The first part ofthe analysis performed in subsection 6.1 of [12] goes through in the present case, the onlysubstantial difference being that the relation between the quaternionic vielbein u, v and the

31

Sp(1) connection ωx is here slightly more involved than (5.4); this leads to a modificationof the equations arising from the hyperino variation. After a few manipulations, we arriveat the following N = 1 AdS vacuum condition for our coset reductions (both ± signs areallowed by susy),

−[

(ImN )−1 AB +3ρ2 + c

ρ2eKXAXB

]

P1B ± i(ImN )−1 AB

(

P3B −NBCP3C

)

= 0 , (B.2)

the (string frame) AdS cosmological constant being given by

Λ = −3

2eK |qAXA|2 . (B.3)

We now solve the susy condition in the Nearly Kahler limit. As in subsection 6.2, wedefine q ≡ ∑

a qa, we rename the only non-vanishing fluxes as e0 → e , m0 → m, andwe set va = v and ba = b for all a. Separating (B.2) into real and imaginary parts, andrecalling (A.6) for K, as well as (A.8), (A.9) for N , we obtain the four real equations

b = ± 4ρ

5ρ2 − c

mIv3

q, b2 =

ρ2 + 3c

15ρ2 − 3cv2

−be +(

b2 +v2

3

)

qξ + mI(b4 + v2b2) = 0 , −e + bqξ + Imb3 ± 3ρ2 + c

4ρqv = 0 .

This system of equations is solved by

v = vT5x − c

(5x + 3c)x1

2

, b = bT

[

(x + 3c)3

x(x − c/5)

]1

4

, ξ = ξT

[

x(x + 3c)

x − c/5

]1

2

, ρ2 = ρ2Tx ,

(B.4)where by vT , bT , ξT , ρ2

T we denote the tree level values (6.3) (recall that at tree level ρ2 isidentified with e−2ϕ). We have also defined c=ρ−2

T c (note that this depends on the values

of the fluxes appearing in ρ2T ∼ (Ime2)

2

3 q−2), while x is the unique positive solution tothe equation

(5x + 3c)4(x + 3c)x − 5(5x − c)3 = 0 , (B.5)

and can easily be determined numerically. The cosmological constant (B.3) here reads

Λ = − q2

5IvT

(x + 3c/5)x3

2

(x− c/5)2.

The tree level result is recovered by taking c = 0, in which case (B.5) is solved by x = 1.We have also checked that (B.4), (B.5) extremize the all loop scalar potential (5.8).

We conclude that string loops preserve the main outcome of the tree level analysis: for anychoice of the fluxes e, m, there exists a unique Nearly Kahler supersymmetric solution.This is however shifted from the tree level position as shown in (B.4). It would beinteresting to study the lifting of this result to a 10d framework.

32

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